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Theorem disjabrexf 30248
Description: Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
Hypothesis
Ref Expression
disjabrexf.1 𝑥𝐴
Assertion
Ref Expression
disjabrexf (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjabrexf
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfdisj1 5041 . . . 4 𝑥Disj 𝑥𝐴 𝐵
2 nfcv 2981 . . . . 5 𝑥𝑦
3 disjabrexf.1 . . . . . . . . . . 11 𝑥𝐴
43nfcri 2975 . . . . . . . . . 10 𝑥 𝑖𝐴
5 nfcsb1v 3910 . . . . . . . . . . 11 𝑥𝑖 / 𝑥𝐵
65nfcri 2975 . . . . . . . . . 10 𝑥 𝑗𝑖 / 𝑥𝐵
74, 6nfan 1893 . . . . . . . . 9 𝑥(𝑖𝐴𝑗𝑖 / 𝑥𝐵)
87nfab 2988 . . . . . . . 8 𝑥{𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
98nfuni 4843 . . . . . . 7 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
109nfcsb1 3909 . . . . . 6 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵
1110nfeq1 2997 . . . . 5 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
122, 11nfral 3230 . . . 4 𝑥𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
13 eqeq2 2837 . . . . 5 (𝑦 = 𝐵 → ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
1413raleqbi1dv 3408 . . . 4 (𝑦 = 𝐵 → (∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 ↔ ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
15 vex 3502 . . . . 5 𝑦 ∈ V
1615a1i 11 . . . 4 (Disj 𝑥𝐴 𝐵𝑦 ∈ V)
17 simplll 771 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → Disj 𝑥𝐴 𝐵)
18 simpllr 772 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥𝐴)
19 simprl 767 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑖𝐴)
20 simplr 765 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝐵)
21 simprr 769 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝑖 / 𝑥𝐵)
22 csbeq1a 3900 . . . . . . . . . . . . . 14 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
233, 5, 22disjif2 30246 . . . . . . . . . . . . 13 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑖𝐴) ∧ (𝑗𝐵𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
2417, 18, 19, 20, 21, 23syl122anc 1373 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
25 simpr 485 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
26 simpllr 772 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥𝐴)
2725, 26eqeltrrd 2918 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑖𝐴)
28 simplr 765 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝐵)
2922eleq2d 2902 . . . . . . . . . . . . . . 15 (𝑥 = 𝑖 → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3025, 29syl 17 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3128, 30mpbid 233 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝑖 / 𝑥𝐵)
3227, 31jca 512 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑖𝐴𝑗𝑖 / 𝑥𝐵))
3324, 32impbida 797 . . . . . . . . . . 11 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑥 = 𝑖))
34 equcom 2018 . . . . . . . . . . 11 (𝑥 = 𝑖𝑖 = 𝑥)
3533, 34syl6bb 288 . . . . . . . . . 10 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑖 = 𝑥))
3635abbidv 2889 . . . . . . . . 9 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑖𝑖 = 𝑥})
37 df-sn 4564 . . . . . . . . 9 {𝑥} = {𝑖𝑖 = 𝑥}
3836, 37syl6eqr 2878 . . . . . . . 8 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
3938unieqd 4846 . . . . . . 7 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
40 vex 3502 . . . . . . . 8 𝑥 ∈ V
4140unisn 4852 . . . . . . 7 {𝑥} = 𝑥
4239, 41syl6eq 2876 . . . . . 6 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥)
43 csbeq1 3889 . . . . . . 7 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
44 csbid 3899 . . . . . . 7 𝑥 / 𝑥𝐵 = 𝐵
4543, 44syl6eq 2876 . . . . . 6 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4642, 45syl 17 . . . . 5 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4746ralrimiva 3186 . . . 4 ((Disj 𝑥𝐴 𝐵𝑥𝐴) → ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
481, 12, 14, 16, 47elabreximd 30184 . . 3 ((Disj 𝑥𝐴 𝐵𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → ∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
4948ralrimiva 3186 . 2 (Disj 𝑥𝐴 𝐵 → ∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
50 invdisj 5046 . 2 (∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
5149, 50syl 17 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  {cab 2803  wnfc 2965  wral 3142  wrex 3143  Vcvv 3499  csb 3886  {csn 4563   cuni 4836  Disj wdisj 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-nul 4295  df-sn 4564  df-pr 4566  df-uni 4837  df-disj 5028
This theorem is referenced by:  measvunilem  31357
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